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Abelian threefolds in products of projective spaces

Abelian threefolds in products of projective spaces Abh. Math. Sem. Univ. Hamburg 65 (1995), 113-121 BY CH. BIRKENHAKE 0 Introduction Whereas it is easy to embed abelian surfaces into low dimensional projective spaces this is a much harder problem for abelian threefolds (see [3] or [1]). In this note we investigate the existence of abelian threefolds in P1 x Ps, IP2 x P4 and IP3 x P3. The analogous question for abelian surfaces was studied in [4] and [5]. The main results of this note are a) There are no abelian threefolds in P1 x P5 (see Theorem 1.1) b) Every abelian threefold in IP2 � P4 is of the form E x A with a plane cubic E and an abelian surface A of degree 10 in IP4 (see Theorem 2.1) c) There exist abelian threefolds in P3 x P3. In fact, a threedimensional family of such threefolds is explicitly constructed (see Theorem 3.5). 1 Abelian Threefolds in P1 x P5 The aim of this section is to prove the following theorem Theorem 1.1. There is no abelian threefold in ]P1 x ]P5. For the proof we need the following lemma Lemma 1.2. Let A be an abelian variety of dimension g and ~p : http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

Abelian threefolds in products of projective spaces

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02953318
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Abstract

Abh. Math. Sem. Univ. Hamburg 65 (1995), 113-121 BY CH. BIRKENHAKE 0 Introduction Whereas it is easy to embed abelian surfaces into low dimensional projective spaces this is a much harder problem for abelian threefolds (see [3] or [1]). In this note we investigate the existence of abelian threefolds in P1 x Ps, IP2 x P4 and IP3 x P3. The analogous question for abelian surfaces was studied in [4] and [5]. The main results of this note are a) There are no abelian threefolds in P1 x P5 (see Theorem 1.1) b) Every abelian threefold in IP2 � P4 is of the form E x A with a plane cubic E and an abelian surface A of degree 10 in IP4 (see Theorem 2.1) c) There exist abelian threefolds in P3 x P3. In fact, a threedimensional family of such threefolds is explicitly constructed (see Theorem 3.5). 1 Abelian Threefolds in P1 x P5 The aim of this section is to prove the following theorem Theorem 1.1. There is no abelian threefold in ]P1 x ]P5. For the proof we need the following lemma Lemma 1.2. Let A be an abelian variety of dimension g and ~p :

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Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals

Published: Sep 11, 2008

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